MAFS.912.A-APR.1.1

Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

Clarifications

Algebra 1 - Fluency Recommendations

Fluency in adding, subtracting, and multiplying polynomials supports students throughout their work in algebra, as well as in their symbolic work with functions. Manipulation can be more mindful when it is fluent.

General Information
Subject Area: Mathematics
Grade: 912
Domain-Subdomain: Algebra: Arithmetic with Polynomials & Rational Expressions
Cluster: Level 1: Recall
Cluster: Perform arithmetic operations on polynomials. (Algebra 1 - Major Cluster) -

Clusters should not be sorted from Major to Supporting and then taught in that order. To do so would strip the coherence of the mathematical ideas and miss the opportunity to enhance the major work of the grade with the supporting clusters.

Date Adopted or Revised: 02/14
Content Complexity Rating: Level 1: Recall - More Information
Date of Last Rating: 02/14
Status: State Board Approved
Assessed: Yes
Test Item Specifications

  • Assessment Limits :
    Items set in a real-world context should not result in a nonreal answer if the polynomial is used to solve for the unknown. 

    In items that require addition and subtraction, polynomials are limited to monomials, binomials, and trinomials. The simplified polynomial should contain no more than six terms. 

    Items requiring multiplication of polynomials are limited to a product of: two monomials, a monomial and a binomial, a monomial and a trinomial, two binomials, and a binomial and a trinomial.

  • Calculator :

    Neutral

  • Clarification :
    Students will relate the addition, subtraction, and multiplication of integers to the addition, subtraction, and multiplication of polynomials with integral coefficients through application of the distributive property. 

    Students will apply their understanding of closure to adding, subtracting, and multiplying polynomials with integral coefficients. 

    Students will add, subtract, and multiply polynomials with integral coefficients.

  • Stimulus Attributes :
    Items may be set in a mathematical or real-world context. 

    Items may use function notation.

  • Response Attributes :
    Items may require the student to write the answer in standard form. Items may require the student to recognize equivalent expressions. 

    Items may require the student to rewrite expressions with negative exponents, but items must not require the student to rewrite rational expression as seen in the standard MAFS.912.A-APR.4.7.

Sample Test Items (1)
  • Test Item #: Sample Item 1
  • Question:

    Roxanne wants to test the idea that polynomials are closed under addition. Her work and explanation are shown.

    Drag an expression and phrases to the boxes to complete the statement.

    Add begin mathsize 12px style 3 x to the power of 4 minus 7 x cubed plus 12 x minus 9 end style to ____________________. The sum is closed because  ______________ are ________________ and by definition the sum is a polynomial.

  • Difficulty: N/A
  • Type: GRID: Graphic Response Item Display

Related Courses

This benchmark is part of these courses.
1200310: Algebra 1 (Specifically in versions: 2014 - 2015, 2015 - 2022 (current), 2022 and beyond)
1200320: Algebra 1 Honors (Specifically in versions: 2014 - 2015, 2015 - 2022 (current), 2022 and beyond)
1200330: Algebra 2 (Specifically in versions: 2014 - 2015, 2015 - 2022 (current), 2022 and beyond)
1200340: Algebra 2 Honors (Specifically in versions: 2014 - 2015, 2015 - 2022 (current), 2022 and beyond)
1200380: Algebra 1-B (Specifically in versions: 2014 - 2015, 2015 - 2022 (current), 2022 and beyond)
1200400: Foundational Skills in Mathematics 9-12 (Specifically in versions: 2014 - 2015, 2015 - 2022 (current), 2022 and beyond)
1206330: Analytic Geometry (Specifically in versions: 2014 - 2015 (course terminated))
1200410: Mathematics for College Success (Specifically in versions: 2014 - 2015, 2015 and beyond (current))
1200700: Mathematics for College Algebra (Specifically in versions: 2014 - 2015, 2015 - 2022 (current), 2022 and beyond)
7912070: Access Liberal Arts Mathematics (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2019, 2019 and beyond (current))
7912090: Access Algebra 1B (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2019, 2019 and beyond (current))
1200315: Algebra 1 for Credit Recovery (Specifically in versions: 2014 - 2015, 2015 - 2022 (current), 2022 and beyond)
1200335: Algebra 2 for Credit Recovery (Specifically in versions: 2014 - 2015, 2015 - 2019 (course terminated))
1200385: Algebra 1-B for Credit Recovery (Specifically in versions: 2014 - 2015, 2015 and beyond (current))
1207300: Liberal Arts Mathematics 1 (Specifically in versions: 2014 - 2015, 2015 and beyond (current))
7912075: Access Algebra 1 (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2019, 2019 and beyond (current))
7912095: Access Algebra 2 (Specifically in versions: 2016 - 2018, 2018 - 2019, 2019 and beyond (current))
1200387: Mathematics for Data and Financial Literacy (Specifically in versions: 2016 - 2022 (current), 2022 and beyond)

Related Access Points

Alternate version of this benchmark for students with significant cognitive disabilities.
MAFS.912.A-APR.1.AP.1b: Understand the concepts of combining like terms and closure.
MAFS.912.A-APR.1.AP.1c: Add, subtract, and multiply polynomials and understand how closure applies under these operations.

Related Resources

Vetted resources educators can use to teach the concepts and skills in this benchmark.

Assessments

Sample 4 - High School Algebra 1 State Interim Assessment:

This is a State Interim Assessment for 9th-12th grades.

Type: Assessment

Sample 1 - High School Algebra 1 State Interim Assessment:

This is the State Interim Assessment for high school.

Type: Assessment

Formative Assessments

Subtracting Polynomials:

Students are asked to find the difference of two polynomials and explain if the difference of polynomials will always result in a polynomial.

Type: Formative Assessment

Multiplying Polynomials - 2:

Students are asked to multiply polynomials and explain if the product of two polynomials always results in a polynomial.

Type: Formative Assessment

Multiplying Polynomials - 1:

Students are asked to multiply polynomials and explain if the product of polynomials always results in a polynomial.

Type: Formative Assessment

Adding Polynomials:

Students are asked to find the sum of two polynomials and explain if the sum of polynomials always results in a polynomial.

Type: Formative Assessment

Lesson Plans

Manipulating Polynomials:

This lesson unit is intended to help you assess how well students are able to manipulate and calculate with polynomials. In particular, it aims to identify and help students who have difficulties in switching between visual and algebraic representations of polynomial expressions, performing arithmetic operations on algebraic representations of polynomials, factorizing and expanding appropriately when it helps to make the operations easier.

Type: Lesson Plan

Laying Tiles for a Polynomial Renovation:

In this lesson students will learn how to add and subtract polynomials.

Type: Lesson Plan

Wonka's Golden Ticket! (Polynomial Operations):

This lesson is an introduction to Polynomials. It includes some activities that will help the teacher to assess student's understanding of several concepts (integer addition and subtraction, combining like terms), which are essential to polynomial operations. Through engaging activities involving candy and chocolates, students will learn to add, subtract, and multiply polynomials!

Type: Lesson Plan

Original Student Tutorials

Introduction to Polynomials, Part 2 - Adding and Subtracting:

Learn how to add and subtract polynomials in this online tutorial. You will learn how to combine like terms and then use the distribute property to subtract polynomials.

This is part 2 of a two-part lesson. Click below to open part 1.

Type: Original Student Tutorial

Introduction to Polynomials, Part 1:

Learn how to identify monomials and polynomials and determine their degree in this interactive tutorial.

This is part 1 in a two-part series. Click HERE to open Part 2.

Type: Original Student Tutorial

Student Center Activity

Method to Multiplying Polynomials:

This video will demonstrate how to multiply polynomials.

Type: Student Center Activity

Tutorials

Addition and Subtraction of Polynomials:

This video tutorial shows students: the standard form of a polynomial, how to identify polynomials, how to determine the degree of a polynomial, how to add and subtract polynomials, and how to represent the area of a shape as an addition or subtraction of polynomials.

Type: Tutorial

Subtracting Polynomials with Multiple Variables:

This video explains how to subtract polynomials with multiple variables and reinforces how to distribute a negative number.

Type: Tutorial

Squaring a Binomial:

This video covers squaring a binomial with two variables. Students will be given the area of a square.

Type: Tutorial

Special Products of Binomials:

The video tutorial discusses about two typical polynomial multiplications. First, squaring a binomial and second, product of a sum and difference.

Type: Tutorial

Multiplying Polynomials:

This tutorial will help the learners practice multiplication of polynomials. Learners will understand that when they multiply expressions with more than two terms, they need to make sure each term in the first expression multiplies every term in the second expression.

Type: Tutorial

Multiplying Bionomials:

Binomials are the polynomials with two terms. This tutorial will help the students learn about the multiplication of binomials. In multiplication, we need to make sure that each term in the first set of parenthesis multiplies each term in the second set.

Type: Tutorial

Multiplying Polynomials:

This resource is a step-by-step tutorial on how to multiply polynomials.

Type: Tutorial

Unit/Lesson Sequence

Sample Algebra 1 Curriculum Plan Using CMAP:

This sample Algebra 1 CMAP is a fully customizable resource and curriculum-planning tool that provides a framework for the Algebra 1 Course. The units and standards are customizable and the CMAP allows instructors to add lessons, worksheets, and other resources as needed. This CMAP also includes rows that automatically filter and display Math Formative Assessments System tasks, E-Learning Original Student Tutorials and Perspectives Videos that are aligned to the standards, available on CPALMS.

Learn more about the sample Algebra 1 CMAP, its features and customizability by watching the following video:

 
 
 

Using this CMAP

To view an introduction on the CMAP tool, please click here

To view the CMAP, click on the "Open Resource Page" button above; be sure you are logged in to your iCPALMS account.

To use this CMAP, click on the "Clone" button once the CMAP opens in the "Open Resource Page." Once the CMAP is cloned, you will be able to see it as a class inside your iCPALMS My Planner (CMAPs) app.

To access your My Planner App and the cloned CMAP, click on the iCPALMS tab in the top menu.

All CMAP tutorials can be found within the iCPALMS Planner App or at the following URL: http://www.cpalms.org/support/tutorials_and_informational_videos.aspx 

Type: Unit/Lesson Sequence

Virtual Manipulatives

Adding and Subtracting Polynomials:

This resource will assess students' understanding of addition and subtraction of polynomials.

Type: Virtual Manipulative

Algebra Tiles (Multiplying Binomials):

This virtual manipulative is intended to allow the student to practice multiplication of binomials. The student should understand how to use algebra tiles before using this tool.

Type: Virtual Manipulative

MFAS Formative Assessments

Adding Polynomials:

Students are asked to find the sum of two polynomials and explain if the sum of polynomials always results in a polynomial.

Multiplying Polynomials - 1:

Students are asked to multiply polynomials and explain if the product of polynomials always results in a polynomial.

Multiplying Polynomials - 2:

Students are asked to multiply polynomials and explain if the product of two polynomials always results in a polynomial.

Subtracting Polynomials:

Students are asked to find the difference of two polynomials and explain if the difference of polynomials will always result in a polynomial.

Original Student Tutorials Mathematics - Grades 9-12

Introduction to Polynomials, Part 1:

Learn how to identify monomials and polynomials and determine their degree in this interactive tutorial.

This is part 1 in a two-part series. Click HERE to open Part 2.

Introduction to Polynomials, Part 2 - Adding and Subtracting:

Learn how to add and subtract polynomials in this online tutorial. You will learn how to combine like terms and then use the distribute property to subtract polynomials.

This is part 2 of a two-part lesson. Click below to open part 1.

Student Resources

Vetted resources students can use to learn the concepts and skills in this benchmark.

Original Student Tutorials

Introduction to Polynomials, Part 2 - Adding and Subtracting:

Learn how to add and subtract polynomials in this online tutorial. You will learn how to combine like terms and then use the distribute property to subtract polynomials.

This is part 2 of a two-part lesson. Click below to open part 1.

Type: Original Student Tutorial

Introduction to Polynomials, Part 1:

Learn how to identify monomials and polynomials and determine their degree in this interactive tutorial.

This is part 1 in a two-part series. Click HERE to open Part 2.

Type: Original Student Tutorial

Student Center Activity

Method to Multiplying Polynomials:

This video will demonstrate how to multiply polynomials.

Type: Student Center Activity

Tutorials

Addition and Subtraction of Polynomials:

This video tutorial shows students: the standard form of a polynomial, how to identify polynomials, how to determine the degree of a polynomial, how to add and subtract polynomials, and how to represent the area of a shape as an addition or subtraction of polynomials.

Type: Tutorial

Subtracting Polynomials with Multiple Variables:

This video explains how to subtract polynomials with multiple variables and reinforces how to distribute a negative number.

Type: Tutorial

Squaring a Binomial:

This video covers squaring a binomial with two variables. Students will be given the area of a square.

Type: Tutorial

Special Products of Binomials:

The video tutorial discusses about two typical polynomial multiplications. First, squaring a binomial and second, product of a sum and difference.

Type: Tutorial

Multiplying Polynomials:

This tutorial will help the learners practice multiplication of polynomials. Learners will understand that when they multiply expressions with more than two terms, they need to make sure each term in the first expression multiplies every term in the second expression.

Type: Tutorial

Multiplying Bionomials:

Binomials are the polynomials with two terms. This tutorial will help the students learn about the multiplication of binomials. In multiplication, we need to make sure that each term in the first set of parenthesis multiplies each term in the second set.

Type: Tutorial

Virtual Manipulatives

Adding and Subtracting Polynomials:

This resource will assess students' understanding of addition and subtraction of polynomials.

Type: Virtual Manipulative

Algebra Tiles (Multiplying Binomials):

This virtual manipulative is intended to allow the student to practice multiplication of binomials. The student should understand how to use algebra tiles before using this tool.

Type: Virtual Manipulative

Parent Resources

Vetted resources caregivers can use to help students learn the concepts and skills in this benchmark.

Tutorials

Addition and Subtraction of Polynomials:

This video tutorial shows students: the standard form of a polynomial, how to identify polynomials, how to determine the degree of a polynomial, how to add and subtract polynomials, and how to represent the area of a shape as an addition or subtraction of polynomials.

Type: Tutorial

Special Products of Binomials:

The video tutorial discusses about two typical polynomial multiplications. First, squaring a binomial and second, product of a sum and difference.

Type: Tutorial

Multiplying Polynomials:

This tutorial will help the learners practice multiplication of polynomials. Learners will understand that when they multiply expressions with more than two terms, they need to make sure each term in the first expression multiplies every term in the second expression.

Type: Tutorial

Multiplying Bionomials:

Binomials are the polynomials with two terms. This tutorial will help the students learn about the multiplication of binomials. In multiplication, we need to make sure that each term in the first set of parenthesis multiplies each term in the second set.

Type: Tutorial

Multiplying Polynomials:

This resource is a step-by-step tutorial on how to multiply polynomials.

Type: Tutorial

Virtual Manipulatives

Adding and Subtracting Polynomials:

This resource will assess students' understanding of addition and subtraction of polynomials.

Type: Virtual Manipulative

Algebra Tiles (Multiplying Binomials):

This virtual manipulative is intended to allow the student to practice multiplication of binomials. The student should understand how to use algebra tiles before using this tool.

Type: Virtual Manipulative